Modeling and Simulation of Microdialysis in the Deep

Modeling and Simulation of Microdialysis in the Deep

Brain Structures

Elin Diczfalusy

Linköping Studies in Science and Technology Licentiate Thesis No. 1549

Department of Biomedical Engineering Linköping University

Linköping 2012

 

                     

Modeling and simulation of microdialysis in the deep brain structures   

Linköping Studies in Science and Technology Licentiate Thesis No. 1549  

Copyright © 2012 Elin Diczfalusy, unless otherwise noted.

Department of Biomedical Engineering Linköping University

SE-581 85 Linköping, Sweden

This is a Swedish Licentiate Thesis. The Licentiate degree comprises 120 ECTS credits for postgraduate studies.

ISBN: 978-91-7519-805-7 ISSN: 0280-7971

Printed by LiU-Tryck Linköping, 2012

Abstract 

Microdialysis is a method for monitoring of the local biochemical environment in a region of interest.

The method uses a catheter, mimicking the function of a blood capillary, to sample substances from the surrounding medium through diffusion. A recent application for microdialysis is the sampling of neuroactive substances in the deep brain, or basal ganglia, during deep brain stimulation (DBS) for patients with Parkinson’s disease. The basal ganglia consist of nuclei interconnected by chemical synapses, and it is hypothesized that the levels of neurotransmitter substances around the synapses are affected by DBS treatment. In order to relate the microdialysis data to their anatomical origin and to the effects of DBS, it is suitable to estimate the tissue volume which is sampled during a microdialysis experiment. In this thesis, the maximum tissue volume of influence (TVI

max

) for a microdialysis catheter was simulated and evaluated using the finite element method (FEM), to allow interpretation of biochemical data in relation to anatomical structures.

A FEM model for simulation of the TVI

max

for a microdialysis catheter placed in grey brain matter was set up, using Fick’s law of diffusion. The model was used to investigate the impact of the analyte diffusion coefficient (D), the tissue tortuosity (λ) and the loss rate constant (k) on the size of the TVI

max

by regression analysis. Using relevant parameter intervals, the radius of the TVI

max

of a neurotransmitter was estimated to 0.85 ± 0.25 mm. A microdialysis experiment on calf brain tissue showed agreement with the regression model. A heterogeneous anisotropic FEM model based on diffusion tensor imaging (DTI) showed that the radius of the TVI

max

may vary by up to 0.5 mm as a consequence of local tissue properties, which was reasonable in relation to the 95% confidence interval from the regression estimation. The TVI

max

was simulated and patient-specifically visualized in relation to MRI images for four patients undergoing microdialysis in parallel to DBS. The size of the TVI

max

showed to be relevant in relation to the basal ganglia nuclei, and the obtained microdialysis data indicated that the biochemical response to DBS depends on the catheter position. The simulations of the TVI

max

were combined with patient-specific DBS electric field simulations, for further interpretation of the results in relation to the effects of DBS.

In conclusion, simulations and visualizations of the TVI

max

allowed relating microdialysis data to its

anatomical origin. Detailed knowledge about the parameters affecting the microdialysis sampling

volume is valuable for the current application as well as other applications related to the migration of

analytes in tissue.

Sammanfattning 

Mikrodialys är en metod som används för studera lokala nivåer av biokemiska substanser i ett specifict organ eller struktur. Metoden använder sig av en kateter med ett semipermeabelt membran, över vilket utbyte av substanser sker genom diffusion. Mikrodialys har nyligen använts för att studera nivåer av neurotransmittorer i de djupa hjärnstrukturerna, ävan kallade basala ganglierna, under djup

hjärnstimulering (DBS) för patienter med Parkinsons sjukdom. De basala ganglierna består av ett antal millimeterstora hjärnstrukturer, sammankopplade via biokemiska synapser, och nivåerna av

signalsubstanser runt dessa synapser tros påverkas av DBS. För att relatera mikrodialysmätningarna till dess anatomiska ursprung, och till effekterna av DBS, är det önskvärt att få en uppskattning av den vävnadsvolym som påverkar mätningen från en mikrodialyskateter. Målet med denna

licentiatavhandling har varit att simulera och utvärdera den maximala påverkansvolymen (TVI

max

) för en mikrodialyskateter med hjälp av finita element-metoden (FEM), för att underlätta tolkningen av de biokemiska data som samlats in.

En FEM-modell sattes upp för att simulera TVI

max

för en kateter placerad i grå hjärnvävnad, baserat på Ficks diffusionslag och lämpliga rand- och initialvillkor. Modellen användes för att göra en

regressionsanalys av hur TVI

max

påverkades av analytens diffusionskoefficient (D), hjärnvävnadens tortuositet (λ) och analytens nedbrytningshastighet (k), och radien för TVI

max

för en neurotransmitter uppskattades till 0.85 ± 0.25 mm då fysiologiskt relevanta parameterintervall användes. En experimentell studie av mikrodialys på hjärnvävnad från kalv gav god överensstämmelse med simuleringsresultaten. En heterogen och anisotrop FEM-modell sattes upp med hjälp av diffusionstensordata (DTI), vilket visade att lokala vävnadsegenskaper påverkar diffusionen av analyter i de basala ganglierna med upp till 0.5 mm i enighet med den regressionsmodell som tagits fram. TVI

max

simulerades och visualiserades sedan i relation till MRI-bilder för fyra patienter som genomgått mikrodialys parallellt med DBS. Målområdena för mikrodialysmätningarna visade sig skilja mellan patienterna, och den insamlade mikrodialysdatan indikerade att den biokemiska responsen på DBS berodde på kateterns position. För att ytterligare underlätta tolkningen av resultatet i relation till effekterna av DBS, kombinerades TVI

max

-simuleringarna med simuleringar av det elektriska fältet runt DBS-elektroderna.

Sammanfattningsvis kan simuleringar av TVI

max

vara en hjälp vid den fysiologiska tolkningen av

insamlad mikrodialysdata, vilket underlättar jämförelser mellan patienter. Detaljerad kunskap om de

parametrar som påverkar samplingsvolymen för en mikrodialyskateter är värdefulla både för den

aktuella applikationen, och övriga applikationer relaterade till diffusion av substanser i vävnad.

List of Publications

The following publications, referred to by their Roman numerals, are included in this thesis:

I. Diczfalusy E, Zsigmond P, Dizdar N, Kullman A, Loyd D and Wårdell K (2011) A model for simulation and patient-specific visualization of the tissue volume of influence during brain microdialysis, Medical & Biological Engineering and Computing, vol 49, pp 1459–1469

II. Diczfalusy E, Dizdar N, Zsigmond P, Kullman A, Loyd D and Wårdell K Simulations and visualizations for interpretation of brain microdialysis data during deep brain stimulation, 34th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, San Diego, USA (In Press, 2012)

III. Diczfalusy E, Andersson M, Wårdell K The effect of tissue heterogeneity and anisotropy on microdialysis of the deep brain – a diffusion tensor based simulation study (In Manuscript, 2012)

Related publications:

Diczfalusy E, Dizdar N, Kullman A, Åström M, Zsigmond P, Wårdell K (2010) Biochemical monitoring and simulation of the electric field during deep brain stimulation, XIX Congress of the European Stereotactic and functional neurosurgery (ESSFN), Athens, 22-25 September 2010.

Diczfalusy E, Åström M, Dizdar N, Kullman A, Zsigmond P, Wårdell K (2010) A finite element model for biochemical monitoring in the brain during deep brain stimulation, World Congress of Neurotechnology, Rome, 11-14 October 2010.

Wårdell K, Diczfalusy E, Åström M (2011) Patient-specific modeling and simulation of deep

brain stimulation, In: Studies in Mechanobiology, Tissue Engineering and Biomaterials,

Springer-Verlag, Berlin-Heidelberg, 2011.

Abbreviations  

BBB Blood-brain barrier

CN Caudate nucleus

CSF Cerebrospinal fluid CT Computed tomography DBS Deep brain stimulation DTI Diffusion tensor imaging ECS Extracellular space FEM Finite element method GABA Gamma-aminobutyric acid GP Globus pallidus GPe Globus pallidus externa GPi Globus pallidus interna MRI Magnetic resonance imaging PD Parkinson’s disease

PPN Pedunculopontine tegmental nucleus

r

TVImax

Radius of the maximum tissue volume of influence SN Substantia nigra

SNr Substantia nigra compacta SNr Substantia nigra reticulata STN Subthalamic nucleus

TVI

max

Maximum tissue volume of influence

Physical symbols  

Scalar symbols are written in italic, vector and tensor symbols are written in bold italic.

µ Magnetic permeability (V·s/A·m) C Concentration (mol/L)

c

b

Analyte concentration at microdialysis catheter boundary (mol/L) D Diffusion coefficient (m

²

/s)

D Diffusion tensor Da Dalton (Da)

D

e

Effective diffusion coefficient (m

²

/s) f Frequency (Hz)

J Current density (A/m

²

) k Loss rate constant (s

^-1

) M

r

Relative molecule mass (-) Q Analyte generation term (-) R

²

Coefficient of determination (-)

u Phase velocity of an electromagnetic wave (m/s) v Bulk flow (m/s)

V Electric potential (V) α Volume fraction (-)

ε Electric permittivity (A·s/V·m) λ Tortuosity (-)

λ’ Wavelength of a field in tissue (m) σ Electric conductivity (S/m)

 

  

Table of contents 

1

 

INTRODUCTION ... 1

 

1.1   T

HE BRAIN

... 1  

1.1.1   Neural cells and nerve signals ... 2  

1.1.2   Tissues of the brain ... 4  

1.1.3   The basal ganglia ... 5  

1.1.4   Parkinson´s disease ... 7  

1.2   D

EEP BRAIN STIMULATION

(DBS) ... 7  

1.2.1   Principles of DBS ... 8  

1.2.2   Advantages, disadvantages and limitations ... 10  

1.3   M

ICRODIALYSIS

... 10  

1.3.1   Principles of microdialysis ... 11  

1.3.2   Advantages, disadvantages and limitations ... 12  

1.4   B

RAIN IMAGING TECHNIQUES

... 12  

1.4.1   Magnetic resonance imaging (MRI) ... 12  

1.4.2   Diffusion tensor imaging (DTI) ... 14  

1.4.3   Computed tomography (CT) ... 15  

1.5   T

HE FINITE ELEMENT METHOD

... 15  

1.5.1   Overview of a FEM simulation ... 15  

1.5.2   Advantages, disadvantages and limitations ... 17  

2

 

AIM OF THESIS ... 19

 

3

 

MATERIALS AND METHODS ... 21

 

3.1   C

LINICAL DATA

... 21  

3.1.1   Human subjects ... 21  

3.1.2   Equipment and clinical procedure ... 21  

3.1.3   Image data ... 22  

3.2   P

HYSICS

... 23  

3.2.1   Diffusion of substances in brain tissue ... 23  

3.2.2   Electric currents in tissue ... 24  

3.3   M

ODELS AND SIMULATIONS

... 26  

3.3.1   FEM models ... 26  

3.3.2   Property matrices and model input parameters ... 26  

3.3.3   Mesh and solver parameters ... 29  

3.4   S

TATISTICAL METHODS

... 30  

3.4.1   Factorial design ... 30  

3.4.2   Regression analysis ... 30  

3.4.3   Descriptive statistics ... 31  

4

 

STUDIES AND RESULTS ... 33

 

4.1   D

EFINITION OF THE

TVI

_MAX

(P

APER

I) ... 33  

4.2   E

VALUATION OF THE

TVI

MAX

(P

APER

I

AND

III) ... 34  

4.3   E

X VIVO EVALUATION

(P

APER

I) ... 34  

4.4   P

ATIENT

-

SPECIFIC SIMULATIONS AND VISUALIZATIONS

(P

APER

I

AND

II) ... 35  

4.5   R

ELATION TO CLINICAL DATA

(P

APER

I) ... 37  

5

 

DISCUSSION AND CONCLUSIONS ... 39

 

5.1   D

EFINITION OF THE

TVI

MAX

... 39  

5.2   M

ODEL INPUT PARAMETERS

... 40  

5.3   E

X VIVO EVALUATION

... 41  

5.4   P

ATIENT

-

SPECIFIC SIMULATIONS

... 41  

5.5   C

LINICAL DATA

... 42  

5.6   C

ONCLUSIONS AND FUTURE DIRECTIONS

... 43  

ACKNOWLEDGMENTS ... 45

 

REFERENCES ... 47

 

1

1 Introduction 

Microdialysis is a method for local sampling of biochemical substances. A recent application for microdialysis is the monitoring of neurotransmitter levels in the basal ganglia in relation to deep brain stimulation (DBS). DBS is an electric stimulation technique which is widely used in clinical practice for treatment of movement disorders such as Parkinson´s disease. The mechanisms of DBS are not entirely clear, and the study of neurotransmitter levels in the basal ganglia may therefore contribute to increased knowledge about the underlying biochemical pathways. To relate the microdialysis sampling and electric stimulation to anatomical structures, patient-specific simulations and visualizations based on the finite element method (FEM) can be used. This chapter provides an introduction to the anatomical and technical aspects of microdialysis and DBS, as well as to the imaging techniques and computational methods used in this thesis.

1.1 The brain 

The brain is the body´s main control and integration unit, where signals from different body parts are registered, integrated and processed. It is the centre for thoughts, emotions, behaviour and memory, all achieved through more or less complex neural processes. The brain is a differentiated organ, with a number of regions specialized for different functions.

An adult brain consists of four main parts: the brainstem, the cerebellum, the diencephalon and the cerebrum (Figure 1) [1].

Figure 1. The main parts of the human brain, based on T1-MRI imaging.

2 1.1.1 Neural cells and nerve signals 

The brain is mainly made up of two kinds of cells; neural cells (neurons) and neuroglia.

Neurons are responsible for transmitting information within the brain and to the rest of the nervous system, allowing functions such as sensing, thinking and controlling movements. The basic parts of a neuron are the cell body, the dendrites and the axon, as shown in Figure 2a.

The cell body contains the nucleus of the neuron, as well as other organelles necessary for the maintenance and function of the cell. The dendrites are the input parts of the neuron, handling incoming signals from other nerve cells, while the axon is the output part propagating nerve impulses towards the synaptic cleft and further to the surrounding cells [2]. Many axons are covered by a protecting and insulating layer known as myelin, in order to insulate and increase the speed of the propagating nerve signals. The myelin is provided by neuroglia cells known as Schwann cells and oligodendrocytes. Overall, neuroglia cells have a supporting function, providing both mechanical and chemical protection to the neurons. The name originates from the fact that neuroglia were originally considered as glue that held the nervous system together [1].

Neurons have a resting membrane potential of about -70 mV, due to a small build-up of negative ions in the cytosol of the cell and a build-up of positive charges in the surrounding extracellular fluid. Numerous ion channels exist in the cell membrane, and when some of these channels open – usually triggered by stimuli such as chemical ligands, mechanical disruption or voltage changes – the membrane potential changes. When a threshold value of about -55 mV is reached, a nerve impulse (action potential) is triggered, propagating the nerve signal throughout the axon. Each nerve impulse consists of a depolarizing phase, caused by an opening of sodium channels, a repolarizing phase, during which potassium channels open, and an after-hypopolarization phase after which the resting state of the membrane is restored [1, 2].

Nerve signals are transmitted from one cell to the next through the synapse (Figure 2b), which is a physical space in between neurons. The synapse may be either electric or chemical;

chemical synapses are much more common within the brain [3]. The chemical synapse

consists of a synaptic cleft, a fluid-filled space of 20-50 nm [1], between the presynaptic and

postsynaptic neurons. In response to a nerve impulse in the presynaptic neuron, certain

chemicals, known as neurotransmitters, are released into the synaptic cleft. The

3

neurotransmitters diffuse through the synaptic fluid and binds to receptors in the membrane of the postsynaptic cell, thereby initiating a postsynaptic potential. The postsynaptic potential may be a depolarization or hyperpolarization, depending on the nature of the neurotransmitter and the receptor. Neurotransmitters causing a depolarizing postsynaptic potential are called excitatory, while neurotransmitters that hyperpolarize the membrane are called inhibitory.

Once released, the neurotransmitters which do not bind to a receptor will be quickly removed from the synaptic cleft by either diffusion out of the synaptic cleft, enzymatic degradation or re-uptake of the substance into the cell [3, 4].

Figure 2. a) The basic parts of a neuron b) Neural signal transmission through the synapse

4 1.1.2 Tissues of the brain 

On a tissue level, the brain consists of grey matter, white matter, blood and cerebrospinal fluid (CSF). Grey matter contains neuronal cell bodies, dendrites, unmyelinated axons and neuroglia. Specific organelles of the neuronal cell body, known as Nissl bodies, give grey matter its greyish look. Grey matter makes up the outer rim of the brain, known as the cortex, as well as several cell body clusterings inside the brain called nuclei. White matter contains myelinated axons, and appears white due to the myelin provided by the neuroglia cells. White matter makes up fibre tracts, which are distributed throughout the brain in order to transmit signals to and from the cortex and the grey nuclei. CSF is a clear liquid which circulates around the brain and the spinal cord, as well as in cavities within them. There are four CSF- filled cavities within the brain known as the ventricles; there are two lateral ventricles located within each hemisphere of the cerebellum (as illustrated in Figure 1), a third ventricle located between the right and left halves of the thalamus, and the fourth ventricle which lies between the cerebellum and the brain stem. The CSF is formed in capillaries in the walls of the lateral ventricles and the roof of the third ventricle, flows towards the fourth ventricle, and on to the subarachonid space which surrounds the brain and spinal cord. The subarachnoid space contains finger-like structures called arachnoid villi, where the CSF is reabsorbed into the blood [4, 5]. The CSF circulation provides mechanical and chemical protection to the brain, and allows substance exchange between the brain and the blood in order to provide the brain with nutrients and remove waste products [4].

Blood vessels are present in both white and grey matter, with blood entering the brain mainly from vertebral arteries and arteries on the sides of the head and neck called the internal carotid. Grey matter is more vascularized than white matter, because of its higher cell content resulting in higher oxygen consumption. Brain capillaries are lined with endothelial cells, but lack the micropore system which is characteristic for capillaries in other parts of the circulation. Therefore, the walls of the brain capillaries are referred to as the blood-brain barrier (BBB). The BBB has a very low permeability for most molecules, thereby protecting the brain from harmful substances. Only small, lipid-soluble substances such as oxygen and carbon dioxide are allowed to diffuse freely over the BBB, while the transport of other substances require active transport mechanisms [5].

 

5 1.1.3 The basal ganglia 

The basal ganglia are a group of interconnected grey nuclei embedded within the cerebral hemispheres. These nuclei have important roles in the initiation, regulation and termination of body movements, and are also involved in cognitive, limbic and linguistic functions [1, 6].

The pathophysiology of common movement disorders such as Parkinson´s and Huntington´s diseases are related to basal ganglia dysfunction [6]. The correct term would be the basal nuclei, since ganglia usually refers to groups of cell bodies outside of the brain [7]; the basal ganglia is, however, the term commonly used in neuroscience and will therefore be used throughout this thesis.

Although the definition of the basal ganglia varies in the literature, three interconnected nuclei are generally considered to be its main components: the caudate nucleus (CN), the putamen and the globus pallidus (GP). The GP is divided into an internal part (globus pallidus interna, GPi) and an external part (globus pallidus externa, GPe), separated by a thin white lamina.

The CN and the putamen are together commonly referred to as the striatum, which is often regarded as the main input station for afferent signals to the basal ganglia network from other parts of the brain. Three additional nuclei, namely the subthalamic nucleus (STN), the substantia nigra (SN) and the pedunculopontine tegmental nucleus (PPN) are integral parts of the pathways through the basal ganglia, although they are not formally parts of it [7, 8]. The

Figure 3. Schematic illustration of the basal ganglia nuclei, here shown in a coronal slice of the brain. Courtesy of Alison Martin (www.sketchymedicine.com)

6

basal ganglia network is connected to the cortex and the thalamus, a nuclei positioned within the diencephalon which helps transmit sensory and motor signals between the different parts of the brain [1].

An overview of the basal ganglia and the surrounding structures is shown in Figure 3. As illustrated in Figure 4, the neural pathways of the basal ganglia are either excitatory or inhibitory, depending on the neurotransmitters that are involved in the synaptic signal transmission. Common neurotransmitters of the basal ganglia include acetylcholine and glutamate which are excitatory, gamma aminobutyric acid (GABA) which is inhibitory, and dopamine and serotonin which may be both inhibitory and excitatory depending on the type of receptor they bind to [1, 9]. The basal ganglia network has been modelled both as a hierarchical network and as a more complex, parallel-acting system of positive and negative feedback systems that “sum up” in the cortex [10-12]. The striatum receives neural input from the cortex, the thalamus and the substantia nigra compacta (SNc). Neural signals are then transmitted from the striatum through the basal ganglia network, projecting to the thalamus and then back to the cortex [12]. The signals through the basal ganglia are thought to be transmitted mainly by two neural paths, called the direct and indirect pathways (Figure 4).

Figure 4. The main neural pathways of the basal ganglia network, based on [7, 8]. Excitatory pathways are shown in black, inhibitory pathways in grey. The left image shows the pathways of a healthy brain, while the right image shows the imbalance caused by Parkinson's disease. The neural pathways from the SNc to the striatum are here degenerated, as indicated by thin dashed arrows.

7

The direct pathway goes from the striatum, projects to the internal globus pallidus (GPi) and the substantia nigra reticulata (SNr) which in turn provides output to the thalamus. In the indirect path, signals are projected through the striatum, to the external globus pallidus (GPe), further on to the STN and then to the GPi and the SNr. The direct and indirect pathways are thought to have opposite effects on the GPi/SNr projections, and the balance between the activities of the pathways regulates the cortical output [7].

1.1.4 Parkinson´s disease 

Parkinson´s disease (PD) is one of the world´s most common neurodegenerative disorders, affecting more than 4 millions of people worldwide [13]. The cardinal features of PD relate to motor dysfunctions, including tremor, dystonia, akinesia and postural instability, but the disease may also include psychiatric symptoms such as anxiety, apathy and depression [14].

PD is thought to stem from a combination of environmental and genetic factors, and is difficult to diagnose due to a lack of biomarkers or visible neuroimaging findings. Therefore, PD is diagnosed based on clinical criteria; mainly the presence of parkinsonian symptoms which cannot be related to other causes [15].

Pathologically, PD is mainly characterized by degeneration of dopaminergic neurons extending from the SNc to the striatum. This degeneration causes an imbalance of neurotransmitter substances in the basal ganglia, affecting dopaminergic as well as non- dopaminergic transmission systems [9]. A summary of how the dopamine depletion affects the pathways of the basal ganglia is seen in Figure 4. Currently, there is no known cure for PD; levodopa medication, which acts as dopamine therapy, is commonly used for symptom relief [16, 17]. As the disorder progresses, symptoms may worsen and side disorders may occur as a consequence of increased medication. In such cases, surgical procedures such as radiofrequency lesioning and deep brain stimulation (DBS) can be used as complementary therapies [17].

1.2 Deep brain stimulation (DBS) 

DBS is a technique for electric stimulation of deep brain structures, within or surrounding the

basal ganglia. During the last decades, more than 40 000 patients over the world have

received DBS implants, showing marked improvements for a growing number of movement

disorders [18-20]. The most common DBS application areas include PD, essential tremor and

8

dystonia. DBS has been approved for PD and essential tremor in the US, Canada, Europe and Australia, and for dystonia in the US [21]. With a high variability and reversibility, DBS offers important advantages over therapies such as radiofrequency lesioning and pharmacological treatment [19]. Besides movement disorders, DBS is also being explored as a treatment option for neuropsychiatric disorders such as depression, obsessive-compulsive disorder and Tourette´s syndrome [21].

1.2.1 Principles of DBS 

In DBS, continuous electrical stimulation is delivered to a pre-defined target area within the brain through chronically implanted electrodes (Figure 5), usually containing multiple contacts [22]. Common target sites for treatment of movement disorders are the STN, the GPi and the ventralis intermediate nucleus (Vim) of the thalamus [21]. The electrodes are connected to an electric stimulator, through which the amplitude of the delivered voltage or current, the pulse width and the frequency can be regulated. By targeting different areas within the brain and by varying stimulation parameters and which electrode contacts to use, DBS can be well adapted depending on the patient´s needs [10]. Commercially available DBS electrodes are about 1.3 mm in diameter, with typical DBS settings ranging from 1-5 V for electric potential, 60-200 µs for pulse width and 130-180 Hz for frequency [23].

Figure 5. X-ray image showing DBS electrodes bilaterally implanted in the subthalamic nucleus (STN).

9

The insertion of DBS electrodes is performed using stereotactic surgery, based on pre- operative imaging and planning (Figure 6) [10]. The stereotactic procedure begins by placing a stereotactic frame on the head of the patient. An indicator box is then applied to the frame, in order to provide fiducials to act as a reference system in the images, and anatomical imaging of the patient´s head is performed. Magnetic resonance imaging (MRI) or computed tomography (CT) is used to obtain the images; commonly, MRI is performed a few days before the implantation, and is then fused with a stereotactic CT obtained on the day of surgery [20]. The pre-operative images are used to identify a suitable entry point and a trajectory towards the DBS target area, where blood vessels and other sensitive structures are avoided. The target can be localized either by direct targeting, based on the pre-operative images, or indirect targeting, based on stereotactic atlases and well-defined brain landmarks [22]. The calculation of coordinates for the intervention is usually performed using commercially available stereotactic software, such as SurgiPlan (Elekta Instrument AB, Stockholm, Sweden), iPlan (BrainLab AG, Munich, Germany), Framelink (Medtronic Incorporation, Minneapolis, MN, USA) or STP (Stereotactic Treatment Planning System;

Howmedica Leibinger GmbH, Freiburg, Germany) [20].

For electrode implantation, a stereotactic arc is attached to the head frame to precisely locate the entry point where a burr hole is made [22]. Intraoperative measurements such as microelectrode recording [24], impedance recording [25] and optical methods [26] can be used prior to electrode insertion, to help confirm that the desired target area has been reached.

Figure 6. Stereotactic implantation of DBS electrodes. A stereotactic frame is placed on the head of the patient, and a stereotactic arc is used for the electrode insertion, as seen in the left image (Courtesy of Elekta Instrument AB, Sweden). The middle image shows a preoperative MRI image with stereotactic reference coordinates, seen as bright spots. The right image shows a postoperative stereotactic CT, where the inserted DBS electrodes are visible as bright artefacts.

10

If the patient is awake during the intervention, DBS electrodes can be tested for efficacy and side effects before the electrode is anchored and secured in place. Finally, the pulse generator is connected to the electrode lead and implanted in the clavicular area. This may be performed immediately after surgery, or several days later [22]. Post-operative MRI or CT imaging (Figure 6) is usually performed in order to verify that the intended target area has been reached [20].

1.2.2 Advantages, disadvantages and limitations 

The greatest benefits of DBS are the reversibility of its effect and the ability to adjust stimulation parameters depending on the target area, the condition to be treated and the patient´s needs [10]. The main disadvantages include the invasiveness of the technique, associated with an increased risk of intracranial bleeding and infections, and potential side effects if the electrodes are incorrectly inserted [18]. Hardware-related complications such as electrode breakage, electrode migration and skin erosions around the pulse generator have also been reported [27].

1.3 Microdialysis 

Microdialysis is an invasive method used for monitoring of the local biochemical environment in a region of interest. The technique uses a catheter with a semi-permeable membrane, allowing measurements of practically all substances smaller than the membrane pores as long as a suitable detection technique is available. Microdialysis of the brain, also known as cerebral microdialysis, is today the most common method for sampling neurochemicals in deep brain structures in vivo for clinical research purposes [28].

Cerebral microdialysis is well established as a laboratory tool, and is also being increasingly used for bedside monitoring during neurointensive care [29, 30]. Clinical microdialysis research applications include conditions such as traumatic brain injury, subarachnoid haemorrhage, ischemic stroke, epilepsy and movement disorders. Within the field of movement disorders, microdialysis has been used in parallel to DBS for patients with PD, to study the local neurochemical environment in basal ganglia structures associated with movement [31-34]. Substances commonly sampled by cerebral microdialysis include dopamine, GABA, glutamate, nitric oxide, glucose, lactate and pyruvate [28].

 

11 1.3.1 Principles of microdialysis 

The basic principle of microdialysis (Figure 7) is to mimic the passive function of a blood capillary. The microdialysis catheter consists of an inner tube, usually ranging from 0.1-0.3 mm in diameter, and an outer semipermeable membrane. The catheter is continuously perfused with a saline solution, called the perfusate. Substance exchange will occur over the membrane, using the concentration gradient between the perfusate and the surrounding medium as the main driving force. Following the substance exchange over the membrane, the resulting microdialysis sample (termed dialysate) is transported through an outlet and collected for analysis [35-37].

The content of the dialysate depends on the pore size (cut-off) of the catheter membrane.

Small pore sizes (≤ 20 kDa) allow collection of small molecules such as neurotransmitters, while pore sizes up to 100 kDa are used to collect macromolecules such as proteins [38]. The relation between the concentration of an analyte in the dialysate and the actual analyte concentration of the surrounding extracellular space is called relative recovery, and depends on parameters such as membrane geometry, flow rate, molecular weight, temperature, and

Figure 7. The principle of microdialysis. Perfusate fluid passes through the catheter, and substance exchange occurs over the membrane depending on the chemical composition of the perfusate and surrounding medium. The perfusate may flow either from the inner tube and out, as illustrated here, or in the opposite direction, depending on the composition of the catheter.

Courtesy of CMA Microdialysis AB, Sweden.

12

sample tortuosity. When performing quantitative microdialysis, knowledge of the extraction fraction is crucial in order to draw conclusions about the actual substance concentration in the tissue [39]. When studying the pattern of concentration change of a substance over time, e.g.

in response to a stimulus, the extraction fraction is of less importance.

When sampling small targets such as deep brain nuclei, which are in the millimeter range, the size of the microdialysis catheter is critical in order to to restrict sampling to only the desired area. Furthermore, the exact placement of the microdialysis catheter must be carefully evaluated and confirmed, especially if the data is meant to reflect the physiology of a particular structure. This confirmation is facilitated by the fact that some commercially available catheters contain a marker at the tip, allowing confirmation of placement by CT or X-ray scan [28, 29]. For brain microdialysis, it should be kept in mind that microdialysis catheters do not sample neurochemicals directly at their release site (such as in the synaptic cleft); instead, it detects the compound in the extracellular fluid surrounding the catheter [40].

1.3.2 Advantages, disadvantages and limitations 

Since microdialysis is an invasive technique, there is a considerable risk of tissue injury [39].

Therefore, cerebral microdialysis is usually applied only when intervention of the brain is already being performed for other reasons. There are few reports of significant injuries from microdialysis in the literature, however [28]. A clear limitation of the technique is the limited time resolution; mean values for a defined period are provided rather than real-time data [39].

The local sampling of a microdialysis catheter is usually considered an advantage, when sampling from a specific structure is desired, but this may also be a limitation if a larger structure or the whole brain is to be studied [28].

1.4 Brain imaging techniques 

1.4.1 Magnetic resonance imaging (MRI) 

MRI is an imaging technique which has been used commercially since the 1980’s [41]. MRI

provides a better soft tissue contrast than tissue density based imaging techniques, such as

computed tomography (CT), and is also considered safer since it does not involve any

ionizing radiation. MRI has therefore gained enormous popularity within clinical imaging and

research, and is today considered an essential tool for the diagnosis of brain and nervous

13

system related disorders. It is also being increasingly used for other body parts such as the heart and the liver [41, 42].

In MRI, the positively charged proton of the hydrogen nuclei is being imaged. A proton has a specific minute magnetic field depending on its charge and spin. When a magnetic field is applied the protons will align with respect to the field. When a radio frequency pulse is applied at the Larmor, or resonance, frequency (42.6 MHz/T), the protons will absorb the energy and start to precess around the main axis of the magnetic field.

While releasing the absorbed energy and returning to the original, “relaxed” state the protons will act as small radio transmitters. There is no way to distinguish the signal from each proton but the joint signal of all excited protons is recorded over time. Now by applying small gradients to the magnetic field during the relaxation the Larmor frequency of the protons now acting as radio transmitters will vary over position and time, q. This feature provides the means to estimate the spatial distribution of the protons but not in terms of a conventional image. These so called k-space samples correspond to the Fourier domain and the spatial image is accessed via an inverse Fourier transform.

The relaxation time for a proton can be described as T1 (spin-lattice relaxation) or T2 (spin- spin relaxation). In the relaxation process, the protons will produce a voltage which is recorded by the MRI scanner, thereby providing information for the resulting intensity image [41]. A MRI image can be T1-weighted, T2-weighted or proton density (spin-density) weighted. The final image intensities depends on parameters such as magnetic field strength, pulse sequence and tissue characteristics. Generally, tissues with a high content of free water (such as CSF) appear dark in T1-weighted images and bright in T2- and proton density- weighted images. Different types of MRI images are used clinically depending on the tissue and the disease to be diagnosed [20, 42]; within the area of stereotactic surgery, proton density is preferred for visualization of the GP, while T2 offers a better view of the STN [20].

Examples of T1, T2 and proton density MRI images are shown in Figure 8.

14 1.4.2 Diffusion tensor imaging (DTI) 

DTI is an MRI-based imaging method which aims to describe the three-dimensional diffusion of water molecules within the brain. DTI stems from “plain” diffusion MRI, which is a method for describing the overall diffusivity of water in brain tissue. Diffusion MRI images are obtained by making the MRI signals sensitive to the diffusion process, using a pair of sharp magnetic field gradient pulses. The first pulse is said to “label” each hydrogen nuclei according to its spatial location, the second pulse is introduced shortly after to detect the displacement that occurred between the pulses (this time interval is known as the diffusion time) [43]. A diffusion MRI image shows the statistical displacement distribution of water molecules within each voxel, presented as a scalar value. DTI images are obtained by collecting diffusion MRI images along several gradient directions, thereby allowing the determination of a second-order tensor model to describe the water diffusion [44].

Diffusion is a physical process that is completely independent of the magnetic field from the MRI scanner, so the imaging does not interfere with the natural diffusion occurring in the tissue. DTI is a powerful tool for exploring the local tissue structure, and is used in clinical research for applications such as white matter tracing [45, 46], detection of brain ischemia [44] and prediction of drug delivery to the brain [47, 48]. It has also been suggested that DTI data is directly related to the electric conductivity of brain tissue [49], and DTI images have therefore been used to simulate the electric potential distribution in the brain during DBS [50- 52].

Figure 8. MRI and CT images of the brain. a) T2-weighted MRI b) T1-weighted MRI c) Proton density weighted MRI (Courtesy of Unit of Functional Neurosurgery, Queen Square , London) d) CT.

15 1.4.3 Computed tomography (CT) 

CT was introduced clinically in 1972, and was the first conventionally used radiological imaging technique that provided computed, digital images. In CT, an X-ray radiation source is rotated around an axis, producing beams with an intensity which is recorded by a detector on the opposite side of the rotation axis. The intensity is measured either in single points or continuously, resulting in an intensity profile. By using inverse transformation, the resulting image can be extracted from the intensity profile. The CT intensities corresponding to different materials for a specific scan are determined depending on their density or attenuation value in relation to water, which are expressed in Hounsfield units (HU) [53]. An example of a brain CT image is included in Figure 8, where bone, having a high HU, appears bright in relation to the soft tissues of the brain.

Clinically, CT is widely used for organs such as the lung, breast, stomach and brain, although criticized for involving large radiation doses [54]. For applications such as DBS implantation, where the exact anatomy of the brain is crucial, CT and MRI are often merged in order to provide a detailed representation of the brain. MRI is considered superior for visualization of soft tissue structures, while CT is is less susceptible to distortions from magnetic field inhomogeneities [20, 22].

1.5 The finite element method 

The finite element method (FEM) is a numerical method used to find approximate solutions to the distribution of field variables within a defined system [55]. The method was originally developed for static structural analysis, but has later been extended to cover domains such as fluid flow analysis, thermal analysis and electric analysis. In recent years, with an increasing understanding of medical and biological systems, FEM has gained popularity for biomedical applications [56]. Topics such as electric field distributions in the brain [57-59], hemodynamics [60, 61], intracranial drug delivery [48, 62], and brain lesioning [63, 64] have all been explored using FEM models.

1.5.1 Overview of a FEM simulation 

The first step of any FEM simulation is the definition of the system to be analysed; the exact

geometry, the material properties or environmental effects and the boundary conditions are all

included in this concept. The next step is the discretization, where the system is divided into a

16

finite number of elements which are connected to each other by nodes located at the element edges or corners. The elements may have different shapes, such as triangular, quadratic or cubic (Figure 9). Elements with straight sides are called linear elements, while elements with curved edges are called higher-order elements and are modelled by introducing midpoint nodes [65].

At each node, the physical quantities of interest are identified, and element equations, also called governing equations, for all elements in the domain are generated. The generation of element equations include determination of the displacement function, which interpolates the displacement field within an element based on the node values. The generated element equations are then assembled to produce the total system equation, based on nodal equilibrium. The boundary conditions, defined in the first step, are then introduced into the system equation [55].

Figure 9. Two examples of meshing (discretization) of a two-dimensional FEM model of a brain electrode. The meshes shown here are have been automatically generated by the FEM software (COMSOL Multiphysics 3.5, Comsol AB, Stockholm, Sweden) with the largest number of elements around corners and edges close to the active electrode. a) Triangular mesh elements b) Quadratic mesh elements.

17

Once the complete system equation has been set up, suitable mathematical methods for solving simultaneous equations are used to find the unknown values, or displacements, at the nodes. Direct or iterative mathematical methods can be used; direct methods are generally considered faster and are suitable for small systems, while iterative methods produce less round-off errors and are preferred for large systems. When the system has been solved, relevant field values can be obtained. Such field values are related to the the physical quantities which have been included in the generation of the system equation; the electric field may be obtained from an electric analysis problem, the heat flux from a heat transfer problem, and so on [55, 66].

1.5.2 Advantages, disadvantages and limitations 

The strength of FEM lies in its ability to estimate field quantities for complex shapes and

systems, for which analytical solutions are often unavailable. The accuracy of the method

depends on the approximations and assumptions which are made about the system; an

increased number of elements and associated nodes will increase the accuracy of the solution,

but will yield a larger number of equations to solve. For complex problems, the number of

nodal equations can easily reach hundreds of thousands or more. The method is therefore

heavily dependent on computer power, and there is a number of commercial software

packages available for finite element analysis [66].

18

19

2 Aim of thesis 

The overall aim of the thesis was to set up patient-specific models, simulations and

visualizations, in order to improve the interpretation of brain microdialysis data in relation to DBS of the STN. Specific aims were to:

 Develop a model for prediction and simulation of the tissue volume of influence (TVI

max

) for a microdialysis catheter, using the finite element method (FEM).

 Investigate the impact of tissue- and substance-related parameters such as tortuosity, diffusion coefficient and loss rate constant on the defined TVI

max

.

 Investigate the effect of tissue heterogeneity and anisotropy on the defined TVI

max

.

 Set up a multiphysics model, in order to simulate and visualize both the TVI

max

for

each microdialysis catheter and the electric field extension around each DBS electrode

for patients undergoing microdialysis in parallel to DBS.

20

21

3 Materials and methods 

3.1 Clinical data 

3.1.1 Human subjects 

In Paper I and Paper II, data from four patients with Parkinson’s disease (aged 56 ± 8) referred for bilateral implantation of DBS electrodes in the STN was used. The patients gave informed written consent for participation in the study (Ethically approved by the Regional Ethics Committee at Linköping University, No. 51-04). In Paper III, images obtained from the author (ED) were used.

3.1.2 Equipment and clinical procedure 

The microdialysis catheters used in the study were model CMA65 (CMA Microdialysis AB, Sweden), with a membrane length of 10 mm, a diameter of 0.4 mm and a molecular cut-off at 20 kDa (Figure 10a). The DBS electrodes were Medtronics Model 3389 (Medtronics Inc.

USA), which has a diameter of 1.27 mm and contains 4 evenly spaced platinum/iridium contacts. The contacts are 1.5 mm in length, and are separated by 0.5 mm (Figure 10b).

Figure 10. Geometrical dimensions of the microdialysis catheters and DBS electrodes used in this thesis. a) CMA65 b) Medtronics 3389.

22

The patients included in Paper I and Paper II each underwent stereotactic implantation of two DBS electrodes in the STN, and three microdialysis catheters in the putamen (right side) and the GPi (left and right side). Stereotactic imaging with a 1.5 T scanner (Achieva, Philips Healthcare, The Netherlands) was performed after placement of the Leksell

^®

Stereotactic System (model G, Elekta Instrument AB, Sweden). Adjacent trans-axial slices of 2 mm thickness with T1 and T2-weighted sequences were used for direct anatomical targeting [67, 68]. The stereotactic images were exported to Leksell

^®

Surgiplan (Elekta Instrument AB, Sweden) for calculation of targets and trajectories. At surgery four burr-holes were placed according to the co-ordinates of the targets. Fluoroscopy images were captured during insertion of the DBS electrodes in order to visualize the electrode positions in relation to the frame and the target coordinates. A stereotactic CT (GE Lightspeed Ultra, GE Healthcare, UK) was done directly after the implantations, to verify the electrode and catheter positions.

Following imaging, each patient was referred to the neuro-intensive care unit where collection of biochemical samples was initiated. Microdialysis data was collected for 72 hours for each patient, after which the catheters were removed.

3.1.3 Image data 

For the patients in Paper I and Paper II, the stereotactic preoperative T2-weighted MRI images and postoperative stereotactic CT images (Section 3.1.2) were used for modeling and visualization of brain tissue in the FEM software. For Paper III, T1-weighted MRI images and DTI images were used, obtained with a 3 T MRI scanner (Ingenia, Philips Healthcare, The Netherlands). All image details are summarized in Table 1.

Table 1. Overview of the patient image data used for the different studies.

Paper I - II I - II III III

Image type T2-MRI CT T1-MRI DTI

Field strength (T) 1.5 - 3 3

Echo time (ms) 20 - 3.6 104.7

Repetition time (ms) 8000 - 8.5 8935

Number of averages 3 - 1 2

Pixel spacing (mm) 0.98 x 0.98 0.55 x 0.55 0.75 x 0.75 1.75 x 1.75

Slice thickness (mm) 2 0.5 0.8 2

23

3.2 Physics 

3.2.1 Diffusion of substances in brain tissue 

On a macroscopic level, diffusion of a substance in a medium can be described by Fick´s second law of diffusion (Equation 1):

 D C 

t

C    



 (1)

C (mol/L) is here the substance concentration C(x,y,z,t), where t is the time (s) and x,y,z (m) are space coordinates. D (m

²

/s) is a parameter known as the substance diffusion coefficient.

The diffusion coefficient describes the ability of an analyte to move through a medium, and depends on the physical properties of the analyte and the solvent. Values of D for different substances can be found experimentally or estimated using a suitable formula [69, 70].

Diffusion in brain tissue takes place mainly in the narrow spaces between the cells, collectively referred to as the extracellular space (ECS). The ECS contains a fluid commonly known as the interstitial fluid, and acts like a porous medium for substances which do not cross the cell membranes. Substances such as neurotransmitters transmit information to surrounding cells by moving through the ECS, a type of communication known as volume transmission [71, 72]. In order to describe the diffusion in the ECS, Equation 1 must be modified in order to take tissue-related parameters into account. Nicholson and Phillips [73]

showed that an unchanged form of the equation could be used to satisfactorily quantify diffusion in the ECS, if averaged variables were used to describe the macroscopic properties of brain tissue. More specifically, two parameters were used in order to set up a volume averaged version of the diffusion equation: the volume fraction α and the tortuosity λ.

Additional parameters were then introduced to account for the release and breakdown of substances as well as bulk flow within the ECS, resulting in Equation 2.

α C f(C) α

C Q λ 2

D t

C       

 



 





 

 v (2)

The tortuosity λ describes the hindrance of diffusion in brain tissue compared to a free

medium, originating from factors such as geometric obstacles and nonspecific interactions

24

with the surrounding cell structures. It is generally assumed that the mean value of λ is about 1.6 for brain tissue, slowing down the diffusion two to three times compared to a free medium. The term D/λ

²

is often denoted D

e

, representing the effective diffusivity of a substance in tissue. D

e

is a second-order tensor for anisotropic tissue, and can be reduced to a scalar, D

e

= tr(D

e

)/3, if the tissue is isotropic [72]. The volume fraction α describes the volume of ECS in relation to the volume of the whole brain. This parameter ranges between 0.15 and 0.30, with a typical value of 0.20. Q/α is a source term representing the release of molecules into the tissue, where Q is a function of space and time, Q(x,y,z,t). The source term may be excluded if the diffusion is driven by substances applied to a surface of the brain; in that case, the substance release can be accounted for through boundary conditions [71]. The term v ·  C accounts for bulk flow, where v is a velocity vector and the velocities are functions of space and time. The existence and amount of bulk flow in brain tissue is debateable, and it is hypothesized that the bulk flow is restricted to the perivascular spaces rather than the entire ECS [74]. f(C)/α describes the rate of analyte loss from the ECS due to factors such as uptake into cells, transport through the blood-brain barrier or enzymatic degradation. The analyte loss is often considered to be proportional to local concentration, so that f(C) = kαC where k is a loss rate constant (s

^-1

) [71, 72].

In order to assure that the local medium properties are averaged, Equation 2 should only be applied to diffusion measurements with distances exceeding 100 µm [75]. When using the equation to solve a diffusion problem, it must be combined with suitable assumptions, initial conditions and boundary conditions.

3.2.2 Electric currents in tissue 

The electric properties of tissues determine the pathways of current flow through the body,

and are therefore of great importance for biomedical applications. Since tissue is highly

heterogeneous on a microscopic level, containing a variety of cell shapes, cell distributions

and extracellular properties which are difficult to mimic in detail, a macroscopic approach is

usually adopted [76]. On a macroscopic level, a material is described as having electric

permittivity, ε (A·s/V·m), and conductivity, σ (S/m). The permittivity describes the ability of

the material to store charge, while the conductivity describes the ability to conduct electric

currents. Tissue contains numerous insulating cell membranes, which work effectively as

capacitors when low-frequency current is applied to the tissue. Therefore, the tissue

25

conductivity is low for low-frequency current and increases with higher frequencies, while the opposite is true for the permittivity [77].

In an electromagnetic sense, the wavelength (m) of a field in a medium is defined as [78]:

εμ f

1 f

λ

^'

 u  ₍₃₎

Here u is the phase velocity of an electromagnetic wave (m/s), µ is the magnetic permeability of the material (V·s/A·m) and f is the frequency (Hz). For typical DBS frequencies of 130-180 Hz, the value of ε is generally less than 1·10

^-5

A·s/V·m [79], giving wavelengths above the meter range. When the wavelength of the applied field is much larger than the region of interest, an electrostatic approximation can be used to describe the electric potential distribution, and the equation of continuity for steady currents can be used [78, 80]:

 σ  V   0









 J (4)

J is the current density (A/m

²

) and V is the electric potential (V). Equation 4 has been used for simulations of DBS [57, 81], modelling the brain as a heterogeneous medium with frequency- dependent electric conductivities. Electric conductivities for the different brain tissue types at some common DBS frequencies are summarized in Table 2. It is seen that cell-containing tissue types, i.e. grey and white matter, have low, frequency-dependent conductivities, while the conductivities of CSF and blood are higher and independent of the applied frequency.

Tissue type Electric conductivity (S/m)

f = 60 Hz f = 130 Hz f = 200 Hz

Grey matter 0.081 0.092 0.094

White matter 0.055 0.059 0.060

Blood 0.70 0.70 0.70

CSF 2.0 2.0 2.0

 

Table 2. Electric conductivities of different brain tissue types at three common DBS frequencies [77].

26

3.3 Models and simulations 

3.3.1 FEM models 

In this thesis, commercial FEM software (Comsol Multiphysics 3.5, Comsol AB, Sweden) was used to simulate analyte diffusion and electric currents in the brain. For the diffusion simulations, Equation 2 was used as the governing equation, with D, λ and k as user-defined input parameters. Geometric microdialysis catheter models based on CMA65 (10 mm length, 0.4 mm diameter; for details see Section 3.1.2), surrounded by a brain tissue domain, were set up. Equation 4 was used as governing equation for the electric simulations, with σ as input parameter. DBS electrode models based on Medtronics 3389 (1.27 mm diameter; for details see Section 3.1.2) were created and positioned in the brain tissue domain. The boundary and initial conditions for the simulations are summarized in Figure 11.

For statistical evaluation of the TVI

max

(Paper I), a FEM model was set up in an axi- symmetric co-ordinate system. The FEM model included a brain tissue domain modelled as a cylinder, with a height of 20 mm and radius of 10 mm. All tissue and analyte related properties were varied within physiologically relevant intervals, as described in Section 3.3.2.

For simulation and visualization of the TVI

max

and the DBS electric field in relation to patient anatomy (Paper I-III), patient-specific FEM models were set up in a three-dimensional Cartesian co-ordinate system. In each model, the geometry of the surrounding brain tissue was represented with a rectangle, 60x40x40 mm for Paper I and Paper II and 80x80x55 mm for Paper III. The dimensions of the brain tissue domains were set in order to include all basal ganglia structures of interest, i.e. the putamen, the GP, the internal capsule and the STN. The positioning of the electrode and catheter models and the assignment of input parameters to each FEM model was based on the intensity and property matrices described in Section 3.3.2.

3.3.2 Property matrices and model input parameters 

MATLAB-based software developed by our group (ELMA1.0 [57, 82]) was used in order to

set up image-based intensity and property matrices to be applied to the brain tissue domain for

the FEM simulations. The software was used to relate the pre- and postoperative data sets to

each other, segment and classify the patient images and assign physical properties such as

electric conductivity and diffusivity to each voxel. Each type of intensity and property matrix

27

is described below, together with the specific input parameters used for each FEM model. An overview of all intensity and property matrices is provided in Figure 12.

Intensity matrix for electrode and catheter positioning

For the patients in Paper I-II, the postoperative stereotactic CT images were used to create an intensity matrix for placement of the modeled electrodes and catheters at their true patient- specific positions. The positioning was made possible due to image artefacts originating from the DBS electrodes and the gold markers in the tips of the microdialysis catheters.

Calculations were performed, based on the frame fiducial co-ordinates in the pre-and postoperative images, for transfer of the electrode and catheter positions to the MRI intensity matrix.

Intensity matrix for tissue visualization

For all patient-specific simulations (Paper I-III), the preoperative stereotactic MRI images were used to create patient-specific intensity matrices for visualization of patient anatomy in the FEM software. Each voxel was represented in the simulation software with an intensity value, interpolated from the original MRI image.

Figure 11. Initial and boundary conditions for the simulation of a) the TVImax and b) the DBS electric fields. Governing equations are denoted in bold, boundary conditions in italic and initial conditions in grey. In a), C is the analyte concentration (nmol/L), D is the diffusion coefficient (cm²/s), λ is the tissue tortuosity, k is the clearance constant (s^-1) and cb is the analyte concentration at the catheter boundary. In b), J is the current density (A m^-2), V is the electric potential (V), V0 is the electric potential at the active electrode contact(s) and σ is the tissue conductivity (S m^-1).

28 Diffusivity matrix

For the diffusion simulations, a matrix was set up to represent the effective diffusivity in each voxel of the brain tissue model. In Paper I and II, the target areas for the catheters were the putamen and the GPi, which were considered to consist of isotropic, grey matter. Therefore, the diffusion in each voxel was represented by a scalar value corresponding to the effective diffusion coefficient of the substance of interest, D

e

= D/λ

²

. For Paper III, the diffusion in each voxel was represented by a second-order tensor D

e

, derived from the DTI data. In both cases, the loss rate constant k for the brain tissue domain was defined in the simulation software.

For the axi-symmetric model in Paper I, which was used for evaluation of the TVI

max

, the three input parameter values (D, λ, k) were varied within physiologically relevant intervals, and a new matrix was set up for each value combination. The value of D was varied in steps between 4·10

^-6

– 1·10

^-5

cm

²

/s to represent all analytes of interest, the values of λwere considered normally distributed with a mean of 1.59 and a standard deviation of 0.096 [71]

and the values of k were assumed to be evenly distributed on the interval 0.003 – 0.010 s

^-1

[75, 83, 84]. For the three-dimensional FEM models in Paper I and Paper II, D was set to 7.5 · 10

^-6

cm

²

/s, λ was set to 1.6 and k was set to 0.0065 s

^-1

, representing average values for the tissue and analytes of interest [71, 75, 83, 84]. For the DTI-based simulations in Paper III, the mean effective diffusivity of each dataset was set to 1.95 · 10

^-6

, 2.92 · 10

^-6

and 3.51 · 10

^-6

cm

²

/s, respectively, to represent diffusion of substances with D = 5, 7.5 and 9 · 10

^-6

cm

²

/s in a tissue medium with λ = 1.6. The value of k was set to 0.0065 s

^-1

.

Electric conductivity matrix

For the electric field simulations (Paper II), intensity-based segmentation of the MRI images was used for identification of different tissue types, and each tissue type was assigned a frequency-dependent value of σ based on Audreccetti’s online database [79]. Some voxels may contain more than one tissue type, resulting in partial volume effects. Therefore, a linear interpolation function was used to assign approximated conductivity values to such voxels.

This process has been described in detail by Åström et al [57]. For all patients in Paper II, a

DBS frequency of 130 Hz was used, corresponding to σ values of 0.09 S/m for grey matter,

0.06 S/m for white matter, 0.70 S/m for blood and 2.0 S/m for CSF [79].

29 3.3.3 Mesh and solver parameters 

All meshes were automatically generated by the FEM software, with the highest mesh density around corners and edges close to the electrode and catheter models. To preserve the spatial resolution of the MRI and DTI data, the maximum element length was set to 1 mm in Paper I and II and to 1.5 mm in Paper III. The axi-symmetric FEM model (Paper I) was divided into about 3000 triangular mesh elements, while the three-dimensional models consisted of about 1,000,000 (Paper I-II) and 1,500,000 (Paper III) tetrahedral mesh elements, respectively. All models were solved using Comsol Multiphysics’ iterative system solver GMRES with the preconditioner Incomplete LU.

 

Figure 12. Overview of the intensity and property matrices used to represent the brain tissue domain for the FEM simulations.